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1 ..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1

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5 1 1.1 f(x) S(x) = + ( n cos nx + b n sin nx) n = 1 b n = 1 n=1 f(x) cos nx dx f(x) sin nx dx f(x) =S(x) f(x) =x ( <x ) f(x) S(x) = (sin x 1 sin x + 1 sin 3x ). 3 x = S() =. f() = f() S() [, b] ={x : x b} () f(x) [, b] [, b] x 1,...,x 5

6 6 lim f(x j + ε) =f(x j +) ε + lim f(x j ε) =f(x j ) ε (1) () (1) f(x) =1( x ), f(x) =(<x ) f(x) [,] () f(x) =1/x (x ), f(x) =(x =) f(x) [ 1, 1] 1.1. ( ) f(x) [, b] [, b] f (x) [, b] 1.1. (1) () (1) f(x) = x ( 1 x ), f(x) =x ( <x 1) f(x) =( x ), f(x) =1(<x ) () f(x) = x ( 1 x ), f(x) = x ( <x 1) x = ( ) f(x) [, b] x 1,...,x b ( x1 x b ) f(x)dx := f(x) dx. x 1 + x f(x) =x ( x ), f(x)dx f(x) = ( <x ) f(x) f(x)dx = xdx + + dx.

7 1.1.1 () f(x) [, b] lim b b lim ( ) b lim f(x) cos( x) dx = lim f(x) cos x dx =, f(x) sin x dx =. b f(x) sin( x) dx = 7 c <c<b. = b c = [ f(x) = I := f(x) cos xdx f(x) cos xdx + sin x]c 1 b c c f(x) cos xdx f (x) sin xdx b +[ f(x) sin x]b c+ 1 c f(c ) sin (c ) f() f(c ) sin (c ) f() f (x) sin xdx sin 1 c sin 1 f (x) sin xdx + c f(x) M, f (x) M 1, x [, b] f (x) sin xdx I f(c ) sin (c ) + f() sin + 1 M + 1 c M 1 dx,. c f (x) sin x dx

8 ( ) f(x) [,] x f(x) = x f(x +)+f(x ) f(x) =S(x) := + ( n cos nx + b n sin nx) n=1 n,b n f(x) (1) f(x) x f(x) S(x) f(x) = + ( n cos nx + b n sin nx). n=1 () f(x) (1) f(x) =x ( x ) [,] x f(x) f(x) = 4 (cos x cos 3x + 1 cos 5x + ). 5 x = =f() = 4 ( ). 8 = () f(x) =x ( x ) [,] x = n 1 subsection 1.3 f(x) = (sin x 1 sin x + 1 sin 3x ). 3

9 9 x = f() = f() = S() x = / f = f( ) = (sin 1 sin sin 3 ). 4 = f(x) =x ( x ) x = x =, x = 1.1. S N (x) := N + ( n cos nx + b n sin nx) n=1 x S N (x) S(x) (N ) S(x) = + ( n cos nx + b n sin nx) n=1 f n = 1 b n = 1 f(x) cos nxdx, f(x) sin nxdx, S N (x) f(x) = f(x +)+f(x ), N.

10 1 S n (x) = 1 1 f(y)dy {( 1 + =1 = 1 { 1 + = 1 { 1 ) ( 1 f(y) cos ydy cos x + =1 } (cos y cos x + sin y sin x) f(y)dy } + cos (y x) f(y)dy. =1 ) } f(y) sin ydy sin x D n (t) := 1 + cos t = (1) D n ( t) =D n (t) : even function () D n (t +) =D n (t) :-period S n (x) = 1 D n (y x)f(y)dy ( ) cos y = eiy + e iy, i = 1. sin y = eiy e iy i D n (y) = 1 + e iy + e iy =1 = ( e iy + e iy ) = 1 =1 e iy = 1 =1 (e iy )

11 11 = 1 e iny (1 (e iy ) n+1 ) 1 e iy = 1 e iny i y (1 (e iy ) n+1 ) e i y e i y e i y = 1 (e i(n+ 1 )y e i(n+ 1 )y )/i (e i y e i y )/i = sin(n + 1)y sin y D n (y) = sin (n+1)y sin y. D n (y)dy = = + =. ( 1 + cos y)dy =1 =1 cos ydy D n (y)dy =1. 1 = = 1 = 1 = / / sin(n +1) y sin y dy sin(n +1) y sin y dy sin(n +1)y sin y dy y = y sin(n +1)y sin y dy.

12 / sin(n +1)y dy =1. sin y S n (x) S n (x) = 1 D n (y x)f(y)dy. D n (y x)f(y) y -period S n (x) = 1 +x +x y x. S n (x) = x +x +x x D n (y x)f(y)dy. sin(n +1)(y x) f(y) sin y x dy sin(n +1)(y x) f(y) sin y x dy =: I 1 (x)+i (x). I 1 (x) = 1 I (x) = 1 x +x +x x f(y) f(y) (n+1)(y x) sin sin y x (n+1)(y x) sin sin y x dy dy I 1 y x = t y t t = s I 1 (x) = 1 = 1 I 1 (x) = 1 = 1 / / y x y x f(x t) f(x t) f(x s) f(x t) (n+1)( t) sin sin t ( dt) (n+1)t sin dt. sin(n +1)s ds sin s sin(n +1)t dt.

13 13 I y x = t y t t = s I (x) = 1 I (x) = 1 S n (x) = 1 = = 1 / / / / f(x + t) f(x +s) f(x +t) (n+1)t sin dt. (f(x +t)+f(x t)) f(x +t)+f(x t) sin(n +1)s ds sin s sin(n +1)t dt. sin(n +1)t dt sin(n +1)t dt / f(x) = sin(n +1)t dt =1 / f(x) sin(n +1)t dt. S n (x) f(x) = / ( f(x +t)+f(x t) ) f(x) sin(n +1)t dt. δ > S n (x) f(x) S n (x) f(x) = δ ( ) f(x +t)+f(x t) f(x) sin(n +1)t dt + / ( ) f(x +t)+f(x t) f(x) sin(n +1)t dt. δ

14 14 ϕ(t) := δ f(x +t)+f(x t) ϕ(t) t t f(x). sin(n +1)tdt ϕ [,δ] ϕ() =. ϕ(t) t = ϕ (c) : <c<t., n δ / δ ϕ(t) sin(n +1)tdt ϕ(t) f 1 / δ ϕ(t) sin(n +1)tdt, n. 1. [,] f(x) n = 1 b n = 1 f(x) cos nx dx, (n ), f(x) sin nx dx, (n 1)

15 15 f(x) T N (x) = c N + (c n cos nx + d n sin nx) f(x) c n,d n n=1 f(x) c = 1 f(x)e ix dx γ e ix f(x) γ f(x) γ e ix dx ( ) 1..1 f [,] n 1 γ = c ( =, ±1, ±,...,±n) 3 1 f(x) γ e ix dx = 1 ( f(x) γ e )(f(x) ix = 1 { f(x) f(x) γ e ix f(x) =( ). ) γ e ix dx γ e ix + γ e ix γ e ix } dx 3 n γ c ( ) n γ n n +1 γ

16 16 c = 1 c = 1 f(x)e ix dx f(x)e ix dx ( ) = 1 f(x) dx =( ). (γ c + γ c )+ 1 γ e ix γ l e ilx dx l= n l = l [ e ix e ilx 1 dx = i( l) ei( l)x e ix e ilx dx = ] dx =. =, ( ) = 1 f(x) dx = 1 f(x) dx = 1 f(x) dx = 1 f(x) dx (γ c + γ c )+ c c + c + c + γ {(c γ )c γ c } + (c γ )(c γ ) c γ. γ γ 1..1 ( ) f [,] 1 f(x) dx c =

17 17 1 f(x) c e ix dx = 1 f(x) dx c. 1 f(x) dx c. n f(x) 1 f(x) dx <. n c n n c c 1 f(x) dx. = 1.3 f [,] R R f(x) x f(x) = d 1 c = 1 f(x)e ix dx, =, ±1, ±,... f(x ) + f(x +) = c e ix, <x< =

18 18 S n (x) = c e ix f [,] R R S n f d 1 (1 ) f() =f() g f ( +), x = f (x), x (,d 1 ) (d 1,) g(x) = f (d 1 ), x = d 1 f ( ), x = g [,] g c c = 1 = 1 = 1 =( ). { d1 g(x)e ix dx f (x)e ix dx + d 1 { [f(x)e ix ] d 1 + i d1 f() =f(), e i = e i ( ) = i } f (x)e ix dx f(x)e ix dx + [ f(x)e ix] } d 1 + i f(x)e ix dx d 1 f(x)e ix dx = ic. c f c = ic.

19 19 1 g(x) dx 1.3. = c <. = c = = c. c = c 1 c + 1. c c + 1 ( c = c ). = c = c + =1( c + c ) c + c = c + =1 1 g(x) dx + =1 1 <. S n (x) f(x) c e ix = c (x ) =n+1 =n

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

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